Reservoir simulation is widely used in the oil industry for forecasting the behavior of fluid flow in a subterranean reservoir. Reservoir simulation is performed using reservoir simulation models, which are generally non-unique due to a variety of uncertain input parameters. The uncertainty arises due to scarcity of data, inaccuracy of measured data, and the inherent variability and heterogeneity in the geological formation and fluid properties. As a result, model outputs, such as the forecasted hydrocarbon production, may have significant uncertainty. Uncertainty quantification techniques for analyzing and reducing the uncertainty in reservoir simulation are therefore heavily relied on. For example, quantification can be performed using statistical properties such as the mean, standard deviation, percentiles including P10, P50, P90, or more rigorously, the entire probability density function (PDF). Uncertainty quantification plays a key role in reservoir management as the quality and robustness of business decisions are greatly impacted by the uncertainty estimates of the variables involved in the decision making process.
Monte Carlo simulation is an approach commonly used for uncertainty quantification. In Monte Carlo simulation, a large number of realizations of random inputs are generated and solved to obtain a set of model outputs, which can be further analyzed statistically. Monte Carlo simulation is conceptually straightforward and easy to implement, however, it requires a large computational effort. Due to the large number of model simulations typically needed to obtain statistically accurate results, Monte Carlo simulations are cost prohibitive in most real applications of reservoir simulation, especially for large scale problems.
Experimental design methods associated with different response surface methodologies are widely used alternatives for assessing uncertainties in reservoir production and economic appraisal. Experimental design methods are typically more efficient than Monte Carlo simulation. However, a key disadvantage of experimental design methods is that they do not take into account the full probability distributions of the parameters consistently while creating the response surface. In particular, the full PDFs are only used during post-processing and not used for sampling and design. Furthermore, because all samples are equally weighted for response surface generation, there is an inherent assumption that the distributions of these parameters are uniform. As a result, experimental design methods may not be appropriate when parameter distributions are arbitrary, which is common in real-world applications. However, these limitations are often disregarded and experimental design methods are broadly used in the industry.
Another approach for uncertainty quantification that has recently been introduced to the petroleum industry is the probabilistic collocation method (PCM). This approach has been applied to uncertainty quantification in the context of optimization of petroleum reservoir production and for quantification of uncertainty for flow in porous media in hydrogeology and petroleum engineering. In the probabilistic collocation method, dependent random variables are represented by employing bi-orthogonal polynomial functions, or polynomial chaos expansions, as the bases of the random space. The polynomial chaos expansions (PCEs) are orthogonal to each other and also with respect to the specific PDFs of the input random variables. They are capable of encapsulating the possibly nonlinear relationships between input and output random variables, and therefore can be used as proxies to the simulation model for efficient uncertainty quantification. PCEs have a significant advantage over other proxies or response surfaces as they converge to the true distribution of the output random variable of interest, such as cumulative oil production, as the order of the PCE and number of simulations used to calculate the PCE coefficients is increased. In the PCM method, the coefficients of the PCE are calculated via collocation or regression. Like most response surface methods, the simulator is used as a black-box in PCM, and is thus very easy to implement. There are, however, some issues with PCM that limit the practical applicability of the method to large scale problems. Mainly, the number of simulations required with the PCM approach to create the PCE is directly dependent on the number of terms in the PCE, which increases exponentially with the PCE order and number of random variables.
While the above methods can be used to quantify uncertainty for reservoir simulation, a robust and efficient method is needed for uncertainty quantification that avoids the aforementioned shortcomings, particularly when being applied to large scale reservoir simulation problems.